On the quotient set of the distance set

Iosevich, Alex; Koh, Doowon; Parshall, Hans

doi:10.2140/moscow.2019.8.103

Cited by 14 publications

(20 citation statements)

References 6 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In addition, if we assume that k is odd and q ≡ 3 mod 4, then the condition |E| ≥ Cq 3d− 4 4 would be enough. We remark here that when E = A × A for A ⊂ F n q or E = A 2n for A ⊂ F q , the cardinality of φ(E, E) has been studied in earlier papers by Iosevich, Koh, and Parshall [14], Pham and Suk [24], respectively, with different approaches.…”

Section: Introductionmentioning

confidence: 99%

Mattila--Sjölin type functions: A finite field model

Cheong¹,

Koh²,

Pham³

et al. 2021

Preprint

Self Cite

View full text Add to dashboard Cite

Let φ(x, y) : R d ×R d → R be a function. We say φ is a Mattila-Sjölin type function if for any compact set E ⊂ R d , we have φ(E, E) has non-empty interior whenever dim H (E) > d 2 . The usual distance function, φ(x, y) = |x − y|, is conjectured to be a Mattila-Sjölin type function.In the setting of finite fields F q , this definition is equivalent to the statement that φ(E, E) = F q whenever |E| ≫ q d 2 . The main purpose of this paper is to prove the existence of such functions in the vector space F d q . In light of our results, it seems that one can not expect a function to be Mattila-Sjölin type in all dimensions.

show abstract

Section: Introductionmentioning

confidence: 99%

Mattila--Sjölin type functions: A finite field model

Cheong¹,

Koh²,

Pham³

et al. 2021

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…In a recent work, Iosevich, Koh, and Parshall [9] proved that the exponent d/2 holds for the quotient set of the distance set, which is defined by…”

mentioning

confidence: 99%

“…The statement of their result is as follows. [9]). Let F q be a finite field of order q, and E be a set in F d q .…”

mentioning

confidence: 99%

See 1 more Smart Citation

On the structure of distance sets over prime fields

Pham

Suk

2020

Proc. Amer. Math. Soc.

View full text Add to dashboard Cite

show abstract

“…Sums of the distance sets: In the spirit of the Erdős-Falconer distance problem, Iosevich, Parshal, and the first listed author [34] recently studied a related interesting question: how large does a subset A of F d q need to be such that the quotient set of the distance set covers all elements in the field? They showed that if d ≥ 2 is even and |A| ≫ q d/2 , then we have…”

Section: Three-distance Problems Over Finite Spacesmentioning

confidence: 99%

Extension theorems in vector spaces over finite fields

Koh¹

Self Cite

View full text Add to dashboard Cite

In this paper, we break the exponent (d + 1)/2 for some distance problems in the spaces over arbitrary finite fields. We also obtain new extension theorems for paraboloids and spheres in odd dimensions. Our L 2 → L r extension theorems for paraboloids in odd dimensions improve significantly the exponent obtained by the first listed author. Our L p → L 4 extension theorems for spheres in odd dimensions break the Stein-Tomas result toward L p → L 4 which has stood for more than ten years. It follows from the results for spheres that there exists a different extension phenomenon between spheres and paraboloids in odd dimensions, namely, the L p − L 4 estimates for spheres with primitive radii are much stronger than the optimal results for paraboloids given by Mockenhaupt and Tao. We will provide connections between the Erdős-Falconer distance problem over finite fields and extension conjectures for paraboloids and spheres. In addition, we also consider the problem of product of simplices in F d q and related topics. Our method is a combination of techniques from harmonic analysis over finite fields, methods from spectral graph theory, and some tools from group action theory and algebraic combinatorics. 1128 2107 under the assumption |A| ≪ q 7/6 due to Iosevich et al. [33].

show abstract

On the quotient set of the distance set

Abstract: Let Fq be a finite field of order q. We prove that if d ≥ 2 is even and E ⊂ F d q with |E| ≥ 9q d 2 then 2000 Mathematics Subject Classification. 11T24, 52C17.

Cited by 14 publications

References 6 publications

Mattila--Sjölin type functions: A finite field model

Mattila--Sjölin type functions: A finite field model

On the structure of distance sets over prime fields

Extension theorems in vector spaces over finite fields

Contact Info

Product

Resources

About